Proving Convergence of Averages of a Convergent Sequence

[R.P.F.]

Homework Statement

Show that if (x_n) is a convergent sequence, then the sequence given by the averages

y_n = (x_1+ x_2 +...+ x_n)/n

also converges to the same limit.

Homework Equations

The Attempt at a Solution

I think I need to show that for n >= N, |y_n - lim(x_n)| < e for an arbitrary e > 0. I don't know where t start because y_n involves terms before N.
Any help is appreciated! :)

 

[CompuChip]

I'm not sure that this really works, but here's something you might try...

If you call x the limit of x_n, you can estimate
$$|y_n - x| = \left| \frac{x_1}{n} + \frac{x_2}{n} + \cdots + \frac{x_n}{n} - x \right| \le \frac{1}{n} \left( |x_1 - x| + |x_2 - x| + \cdots |x_n - x| \right)$$

At least then you'll have written everything in terms of |x_i - x| which occurs in the limit definition of x_n.

Another thought that occurred to me while writing this message is that you could take the maximum of the (finitely many) x_i, and write
$$(x_1 + x_2 + \cdots + x_n) / n \le n x_i / n = x_i$$
for some i = 1, 2, ..., n.

 

[Gib Z]

A traditional (brute force in a sense) approach: Since $$x_n \to L$$, for every $$\epsilon>0$$ there exists $$n_0\in \mathbb{N}$$ such that for all $$n>n_0$$ we have $$L-\epsilon < x_n < L + \epsilon$$

Then for $$n>n_0$$ we write:

$$\frac{ x_1 + x_2 + ... + x_n}{n} = \frac{ x_1 + x_2 + ... +x_{n_0}}{n} + \frac{x_{n_0 +1} + x_{n_0 +2} + ... + x_n}{n}$$

For the first term, the numerator adds to some finite number, so we can use the Archimedian property to bound that, and for the second term we can use the inequalities above to bound that.

 

[R.P.F.]

Thank you so much, guys!
May I ask how to use LaTeX on PF? Thanks!

 

[Gib Z]

If you click on the LaTex you can see the code that we typed to display what you see. Also there's a reference to a brief guide underneath the code. You'll pick it up by clicking on other peoples images and learning the basic structure of the code.

 

[R.P.F.]

Thank you, Gib. I know how to write LaTeX code. I'm just wondering how to post it onto PF. Do I just add $$and [\tex]?$$

 

[Gib Z]

Yup just those before and after the code. Or for inline, [ itex] and [ /itex]